DT-operators and decomposability of Voiculescu's circular operator

Abstract

The DT-operators are introduced, one for every pair (μ,c) consisting of a compactly supported Borel probability measure μon the complex plane and a constant c>0. These are operators on Hilbert space that are defined as limits in *-moments of certain upper triangular random matrices. The DT-operators include Voiculescu's circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT-operator is strongly decomposable. We also show that a DT-operator generates a II_1-factor, whose isomorphism class depends only on the number and sizes of atoms of μ. Those DT-operators that are also R-diagonal are identified. For a quasi-nilpotent DT-operator T, we find the distribution of T^*T and a recursion formula for general *-moments of T.